the following equation involves a trigonometric equation in quadratic form. solve the equation on the…

the following equation involves a trigonometric equation in quadratic form. solve the equation on the interval 0,2π).\n2 sin²x = 3 sin x + 5\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\no a. x=\n(type an exact answer in terms of π. use integers or fractions for any numbers in the expression. use a comma\nto separate answers as needed.)\no b. there is no solution.

the following equation involves a trigonometric equation in quadratic form. solve the equation on the interval 0,2π).\n2 sin²x = 3 sin x + 5\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\no a. x=\n(type an exact answer in terms of π. use integers or fractions for any numbers in the expression. use a comma\nto separate answers as needed.)\no b. there is no solution.

Answer

Explanation:

Step1: Rearrange the equation

Move all terms to one side: (2\sin^{2}x - 3\sin x - 5 = 0). Let (t=\sin x), then the equation becomes (2t^{2}-3t - 5=0).

Step2: Solve the quadratic equation

Use the quadratic formula (t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}) for (at^{2}+bt + c = 0). Here (a = 2), (b=-3), (c=-5). First, calculate the discriminant (\Delta=b^{2}-4ac=(-3)^{2}-4\times2\times(-5)=9 + 40=49). Then (t=\frac{3\pm\sqrt{49}}{4}=\frac{3\pm7}{4}). We get (t_1=\frac{3 + 7}{4}=\frac{10}{4}=\frac{5}{2}) and (t_2=\frac{3-7}{4}=\frac{-4}{4}=-1).

Step3: Substitute back (t = \sin x)

Since (-1\leqslant\sin x\leqslant1), (\sin x=\frac{5}{2}) is rejected. For (\sin x=-1), on the interval ([0,2\pi)), (x=\frac{3\pi}{2}).

Answer:

A. (x=\frac{3\pi}{2})